Section 7. Characteristics of Quadratic Functions A QUADRATIC FUNCTION is a function of the form " # $ N# 1 & ;# & 0 Characteristics Include:! Three distinct terms each with its own coefficient:! An x term with coefficient a, where a # 0! An x term with coefficient b! A constant term, c! Note: If any term is missing, the coefficient of that term is 0! The graph of this function is called a parabola! If a > 0, then the parabola opens up; otherwise if a < 0, then the parabola opens down! The y-intercept has coordinates (0, c) Problem 3 WORKED EXAMPLE GRAPHS OF QUADRATIC FUNCTIONS Given the Quadratic Function f (x) x + 4x, complete the table. Identity the coefficients a, b, c a 1, b 4, c Which direction does the parabola open? a 1 which is greater than 0 so parabola opens up What is the y-intercept? c so y-intercept (0, ) Problem 4 MEDIA/CLASS EXAMPLE GRAPHS OF QUADRATIC FUNCTIONS Given the Quadratic Function f(x) x x + 3, complete the table. Identity the coefficients a, b, c Which direction does the parabola open? Why? What is the y-intercept? 38
Problem 5 YOU TRY Graph Quadratic Functions Given the Quadratic Function f(x) x 5, complete the table. Identity the coefficients a, b, c Which direction does the parabola open? Why? What is the y-intercept? Vertex of a quadratic function. Given a quadratic function,!!" # $ N# 1 & ;# & 0: The VERTEX is the lowest or highest point (ordered pair) of the parabola. It is the midpoint between the x-intercepts, if they exist.! To find the input value, identify coefficients a and b then compute b! a! Plug this input value into the function to determine the corresponding output value, (i.e. evaluate!" % e ) 1K! The Vertex is always written as an ordered pair. Vertex % e 1K 5 " % e The axis of symmetry is the vertical line that goes through the Vertex, it divides the parabola in half. 1K Problem 6 WORKED EXAMPLE Quadratic Functions: Vertex/Axis Of Symmetry Given the Quadratic Function f (x) x + 4x, complete the table below. Identity the coefficients a, b, c a 1, b 4, c Determine the coordinates of the Vertex. Input Value b x! a (4)! (1)! Output Value f! (! ) ( ) 4! 8!! 6 + 4(! )! Vertex Ordered Pair: (, 6) 39
Identify the Axis of Symmetry Equation. Axis of Symmetry: x Sketch the Graph Problem 7 MEDIA/CLASS EXAMPLE Quadratic Functions: Vertex/Axis Of Symmetry Given the Quadratic Function f (x) x x + 3, complete the table, generate a graph of the function, and plot/label the vertex and axis of symmetry on the graph. Identity the coefficients a, b, c Determine the coordinates of the Vertex. Identify the Axis of Symmetry Equation. Graph of the function. Plot/label the vertex and axis of symmetry on the graph. 40
Problem 8 YOU TRY Quadratic Functions: Vertex/Axis Of Symmetry Given the Quadratic Function f(x) x 5, complete the table, generate a graph of the function, and plot/label the vertex and axis of symmetry on the graph. Identity the coefficients a, b, c Determine the coordinates of the Vertex. Identify the Axis of Symmetry Equation. Graph of the function. Plot/label the vertex and axis of symmetry on the graph. 41
Problem 9 WORKED EXAMPLE Quadratic Functions: Domain and Range Determine the Domain and Range of the Quadratic Function f (x) x + 4x Domain of f(x): All real numbers. & < x < & or ( &, &) Range of f(x): Since the parabola opens upwards, the vertex (, 6) is the lowest point on the graph. The Range is therefore 6! f (x) < & or [ 6, &) Problem 10 MEDIA/CLASS EXAMPLE Quadratic Functions: Domain and Range Determine the Domain and Range of f(x) x 6. Sketch the graph and label the vertex. 4
Vertex: Domain of f (x): Range of f (x): Problem 11 YOU TRY Quadratic Functions: Domain and Range Determine the Domain and Range of f (x) x 5. Sketch the graph and label the vertex. Vertex: Domain of f (x): Range of f(x): 43
Finding x-intercepts of a Quadratic Function The quadratic function, f (x) ax +bx+c, will have x-intercepts when the graph crosses the x-axis (i.e. when f (x) 0). These points are marked on the graph above as G and H. To find the coordinates of these points, what we are really doing is solving the equation ax +bx+c 0. There are several methods to solve the equations: by factoring, by using the quadratic formula or by completing the square. Problem 1 WORKED EXAMPLE Finding x-intercepts of a Quadratic Function Find the x- intercepts of f (x) x + 4x and plot/label them on the graph. 1. To find the x-intercept, set f(x) 0. That means, solve for x when x + 4x 0.. Equation is not factorable. We will have to either complete the square or use the quadratic formula. If you complete the square: x x + 4x! 0 x + 4x + 4x + 4 + 4 ( x + ) 6 If you use the quadratic formula, first identify: a 1, b 4, c ( x + ) x + 6 6 x + ± 6 x! ± 6 "! 4. 45, 0. 45 44
x! 4 ± (4) (1)! 4(1)(! )! 4 ±! 4 ±! 4 ± 16 + 8 4 4 # 6! 4 ± 6! ± 6 "! 4.45, 0.45 3. The x-intercepts are: ( 4.45, 0) and (0.45, 0) 45
Problem 13 MEDIA/CLASS EXAMPLE Finding x-intercepts of a Quadratic Function Given the Quadratic Function f (x) x x 6, find the x-intercepts and plot/label them on the graph. Problem 14 YOU TRY Finding x-intercepts of a Quadratic Function Given the Quadratic Function f (x) x 3x 5, find the vertex, x-intercepts, y-intercept, another point symmetric to the y-intercept. Plot and label all of these points on the graph. Then find domain and range. Vertex: (, ) y-intercept: (, ) Point Symmetric to y-intercept: (, ) x-intercepts: (, ) and (, ) Domain: Range: 46